Annual Returns


BUSI 721, Fall 2022
JGSB, Rice University

Kerry Back

Annual SPY returns


SPY = S&P 500 ETF

Compounded SPY returns


value of $1 investment with dividends reinvested

Compounded returns on log scale: motivation

  • Let’s look at accumulations from two hypothetical stocks.
    • stock 1: 10% per year
    • stock 1: 2% per year until 2000 and 10% afterwards
  • It will appear that stock 2 did nothing before 2000 and earned a lot less than stock 1 even after 2000.

Plot of the Example

Log (base 10) of accumulation

Map \(y\) tick labels to dollars

Compounded SPY returns on log scale


value of $1 investment with dividends reinvested

Box Plot

  • Box contains 25th percentile through 75th percentile.

  • Median is indicated as a line in the box.

  • Fences extend 1.5 times inter-quartile range from 25th and 75th percentiles or to the most extreme observation if that is closer to the box.

    • inter-quartile range = 75th minus 25th percentile)
  • Points outside the fences are outliers.

    • If you simulate data from a normal distribution, there will typically be very few points outside the fences.

Box and density plots of annual SPY returns

Normal distribution has same mean and std dev as actual

Autocorrelations

  • Autocorrelation is the correlation of a time series with its own lagged values.
  • Autocorrelation at lag 1 tells us whether the current value predicts the next one.
  • For monthly data, autocorrelation might be high at lag 12 (seasonality).

Autocorrelations of annual SPY returns

Does last year’s return predict this year’s?

No, the autocorrelation is almost zero.